for any set \(A \subseteq B\) , where \(\gamma \in [0,1]\)  is called the submodularity ratio of \(f\) . If \(\gamma \ge 1\) , the function \(f\)  is a submodular. Since the standard analysis of the greedy algorithm works by bounding the marginal gains of the optimal elements not selected during the algorithm, judicious application of (REF ) gives guarantees for the greedy algorithm even for arbitrary fixed \(\gamma \) .We note that the definition given here, which is also used in [1]}, [2]}, [3]}, [4]}, [5]}, is slightly adapted from the original definition given in [6]}.
